| L(s) = 1 | − 32·17-s + 428·25-s + 1.24e3·41-s − 476·49-s − 984·73-s + 5.56e3·89-s − 1.20e3·97-s + 6.20e3·113-s + 1.74e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.98e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 0.456·17-s + 3.42·25-s + 4.75·41-s − 1.38·49-s − 1.57·73-s + 6.63·89-s − 1.26·97-s + 5.16·113-s + 1.30·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.63·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.34247456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.34247456\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 214 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 34 p T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 870 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3994 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6550 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4338 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46662 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 59134 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 74410 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 312 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 20186 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 178974 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 226998 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 346246 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 436538 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 343478 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 257070 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 246 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 931870 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 195606 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1392 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 302 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.74051311242895162329225037058, −6.26186390420236299262986898798, −6.10943607056014575740970176993, −6.07468490768829373575220726197, −5.77377340618477275039220994881, −5.62013531321938735239769797177, −5.05403635175349150163910024257, −4.94998026026571007814556191358, −4.86024276948427985340841939099, −4.63306543984598574957539009257, −4.21026620089114995824705675234, −4.18917185355927322950393102587, −4.00402171497041564164038638337, −3.33707929018307299575033019024, −3.15862577324839819390455785523, −3.04778070124184201568645686863, −2.99151796830105605266624666683, −2.38570398996095208070971761606, −2.20643780495507675980572054619, −1.91274880559935090060285910323, −1.68938816589699488749758997620, −1.01512033486156650575088197395, −0.76254989645313024606813594334, −0.72792004225291869272143637714, −0.45700417260991164465714301376,
0.45700417260991164465714301376, 0.72792004225291869272143637714, 0.76254989645313024606813594334, 1.01512033486156650575088197395, 1.68938816589699488749758997620, 1.91274880559935090060285910323, 2.20643780495507675980572054619, 2.38570398996095208070971761606, 2.99151796830105605266624666683, 3.04778070124184201568645686863, 3.15862577324839819390455785523, 3.33707929018307299575033019024, 4.00402171497041564164038638337, 4.18917185355927322950393102587, 4.21026620089114995824705675234, 4.63306543984598574957539009257, 4.86024276948427985340841939099, 4.94998026026571007814556191358, 5.05403635175349150163910024257, 5.62013531321938735239769797177, 5.77377340618477275039220994881, 6.07468490768829373575220726197, 6.10943607056014575740970176993, 6.26186390420236299262986898798, 6.74051311242895162329225037058
